This is another problem I found trying to obtain properties related to finite exchangeable sequences of random variables. Anyway, I am asking the question here because I think the answer could be already well known..
Problem: Let $1=a_0 \ge a_1 \ge \cdots \ge a_k \ge 0$ be $k+1$ fixed real numbers. Is it possible to find a probability measure $\mu$ on $([0,1],\mathscr{B}[0,1])$ such that for all $i=0,1,\ldots,k$ it holds $$ \int_{[0,1]}t^i \mu(\mathrm{d}t)=a_i\,\,? $$
The problem, without the additional assumption of non-increasing moments, is called reduced Hausdorff moment problem, but I didn't find a characterization of the cases when the solution is affirmative [indeed, this is going to be not always affirmative]..